   Chapter 5.2, Problem 18E

Chapter
Section
Textbook Problem

# Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. y = x ,   y = 0 ,   x = 2 ,   x = 4 ; about x = 1

To determine

To find:

The volume of the solid obtained by rotating the region bounded by the given curves about the specified line and sketch the region, the solid and a typical disk or washer.

Explanation

1) Concept:

i. If the cross section is a washer with inner radius rin and outer radius rout, then area of washer is obtained by subtracting the area of the inner disk from the area of the outer disk,

ii. The volume of the solid revolution about the y-axis is,

V= abA(y)dy

2) Given:

3) Calculation:

The region is bounded by y=x,y=0, x=2, x=4 and rotated about the line x=1 as shown below:

From above figure, for 0y2, a cross section of the solid is an annulus with outer radius  4-1=3 and inner radius  2-1=1

So it’scross sectional area is,

=π9-1

A1y=8π

Similarly, for 2y4, a cross section of the solid is an annulus with outer radius is 4-1=3 and inner radius is y-1.

So its area is given by,

=π9-(y2-2y+1)

A2

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